Subgraph Patterns: Network Motifs and Graphlets. Pedro Ribeiro


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1 Subgraph Patterns: Network Motifs and Graphlets Pedro Ribeiro
2 Analyzing Complex Networks We have been talking about extracting information from networks Some possible tasks: General Patterns Ex: scalefree, smallworld Community Detection What groups of nodes are related? Node Classification Importance and function of a certain node? Network Comparison What is the type of the network? 2
3 Building Blocks of Networks Subnetworks, or subgraphs, are the building blocks of networks They have the power to characterize and discriminate networks 3
4 Building Blocks of Networks 4
5 Some Graph Definitions Subgraph: subset of vertices and edges inside another graph Induced subgraph: You must consider all connections between selected nodes Induced Subgraph NonInduced Subgraph 5
6 Some Graph Definitions Graph Isomorphism: Two graphs G and H are isomorphic (G ~ H) when there is a one to one mapping between the nodes of both graphs and there is an edge between two vertices of G if and only if the corresponding vertices in H also form an edge Informally we are seeing if the topology is the same 4 Isomorphic Graphs 6
7 Some Graph Definitions Graph Automorphisms: Set of isomorphisms of a graph into itself Equivalence of Nodes Two nodes are equivalent if there exists an automorphism that maps one into the other This relation partitions a graph into equivalence classes Informally, we are looking at the graph symmetry Example graph automorphisms and equivalence classes 7
8 Some Graph Definitions Graph Matches: A match of a graph H in a larger graph G is a set of nodes of G that induce the respective subgraph H. In other words, a subgraph of G that is isomorphic to H 8
9 Example Application of Subgraphs Consider all possible directed subgraphs of size 3 9
10 Example Application of Subgraphs ZScore measures overrepresentation Different networks present similar fingerprints Image: Adapted from (Milo et al., 2004) 10
11 Network Motifs Milo et al. (2002) came up with the definition of network motifs: recurring, significant patterns of interconnections How to define: Pattern: induced subgraph Recurring: found many times, i.e., high frequency Significant: more frequent than it would be expected in similar networks (same degree sequence) Parameters P, U, D, N control the definition (Milo et al., 2002, used {0.01, 4, 0.1, 1000}) Image: Adapted from (Milo et al., 2004) 11
12 Network Motifs Random Networks Motif Original Network Image: Adapted from (Milo et al., 2002) 12
13 Network Motifs: Example Application Table: Adapted from (Milo et al., 2002) 13
14 Network Motifs: Example Application Image: taken from (Sporns and Kotter, 2004) 14
15 Graphlet Degree Distribution What about a node subgraph metric? The degree distribution is in a way measuring participation in subgraphs of size 2 Can we generalize this? Przulj (2006) came up with the definition of graphlet degree distribution: Where does the node appear in orbits of subgraphs? 15
16 Graphlet Degree Distribution 16
17 Computational Problem In its core, finding motifs and graphlets its all about finding and counting subgraphs. Just knowing if a certain subgraph exists is already an hard computational problem! Subgraph isomorphism is NPcomplete Execution time grows exponentially as the size of the graph or the motif/graphlet increases Feasible motif size is usually small (3 to 8) and network size in the order of hundreds or thousands of nodes 17
18 Subgraph Discovery Research Part of my research is focused on improving efficiency in network motif detection. Scale up! How? Novel data structures for the graphs and subgraphs Novel faster algorithms Sampling techniques Parallel approaches (with different paradigms) 18
19 Previous Approaches Networkcentric approaches: Enumerate all subgraphs and then compute isomorphisms (ex: ESU/Fanmod, Kavosh). Subgraphcentric approaches: Find one subgraph at a time (ex: Grochow and Kellis) 19
20 A setcentric approach Key insight: can we do better looking for a given set of subgraphs? Looking for all does not allow for discarding subgraphs we don't care about (which would save computation time) Looking for one at the time does not allow to reuse results (which would avoid redundant computation) Can we find what is common between subgraphs and use that? Setcentric approach: Find a custom set of subgraphs (maybe one, maybe all, maybe something in between) 20
21 GTries Motivation Sequences and Prefix Trees (tries) Can the concept be extended to graphs? 21
22 GTries Motivation Graphs have common substructure! GTries (etimology: Graph retrieval ) 22
23 The GTrie data structure GTries: (customized) collections of subgraphs Common substructures are identified Information is compressed 23
24 The GTrie data structure GTries: also valid for directed networks 24
25 The GTrie data structure GTries: also valid for colored/labeled networks GTrie 25
26 Creating a GTrie Iterative insertion Start with an empty gtrie 26
27 The Need for a Canonical Form There are different node orderings representing the same subgraph Canonical form for a getting an unique gtries Different canon will give origin to different gtries 27
28 Impact of Canonical Form 28
29 Custom Canonical Form Connectivity Path induces connected subgraph Compressibility More common subtructre, less gtries nodes Constraining As many connections as possible to ancestor nodes (limit possible matches) GTCanon 29
30 GTCanon Example 30
31 Searching with GTries Backtracking Procedure Searching at the same time for several subgraphs Candidates for node 1: {0, 1, 2, 3, 4, 5} Try 0: Match = {0}, Neighb. = {1,3,4} Try 1: Match = {0,1}, Neighb. = {2,3,4,5} Try 2: no edge from 2 to 0! FAIL Try 3: no edge from 3 to 1! FAIL Try 4: Match = {0, 1, 4} FOUND! Try 5: no edge from 5 to 1! FAIL 31
32 Searching with GTries The same subgraph could be found several times due to automorphisms (symmetries) We would not only find {0,1,4} but also: {0, 4, 1} {1, 0, 4} {1, 4, 0} {4, 0, 1} {4, 1, 0} 32
33 Symmetry Breaking Conditions Conditions on node labels Add conditions to respective gtrie path Match only when conditions of at least one descendant are respected Filter conditions to ensure minimum work Ex: transitive property (a<b,a<c,b<c leads to a<b,b<c); assured descendants, only store relevant to node, etc 33
34 Complete GTrie Example All six undirected 4subgraphs 34
35 Complete GTrie Example All thirteen directed 3subgraphs 35
36 Sequential version: some results Set of 12 representative real networks Network Group Directed dolphins social no circuit physical no neural biological yes 297 2, metabolic biological yes 453 2, links social yes 1,490 19, coauthors social no 1,589 2, ppi biological no 2,361 6, odlis semantic yes 2,909 18, power physical no 4,941 6, company social yes 8,497 6, foldoc Semantic yes 13, , internet Physical no 22,963 48, ,390 V(G) 36 E(G) Nr. Neighbours Average Max
37 Sequential version: some results On both directed and undirected graphs gtries were from 1 to 2 orders of magnitude faster than existing state of the art From 10x to 200x Example results for full census of size k (speedup on a set of undirected networks) Network k ESU Kavosh Grochow dolphins circuit coauthors ppi power internet
38 Sequential version: some results On both directed and undirected graphs gtries were from 1 to 2 orders of magnitude faster than existing state of the art From 10x to 200x Example results for full census of size k (speedup on a set of directed networks) Network K ESU Kavosh Grochow neural metabolic links odlis company foldoc
39 Sequential version: some results On both directed and undirected graphs gtries were from 1 to 2 orders of magnitude faster than existing state of the art From 10x to 200x 39
40 Sampling approach Sample some subgraph occurrences Deduce approximate total counting Trade accuracy for speed 40
41 Sampling approach: search tree Backtracking produces search tree 41
42 Sampling approach Original: unbalanced search tree Goal: Uniform sampling Subgraph occurrences are in last tree depth 42
43 Sampling approach Original: unbalanced search tree Goal: Uniform sampling Associate a probability of traversing with each search tree depth 100% 100% 80% 80% 43
44 Sampling Approach Probabilities associated with each depth: {P0, P1,, Pmax} Sampling is uniform Probability of any occurrence is P0 x P1 x x Pmax We can produce unbiased estimator Estimate = (Nr sampled occurences) / (P0 x P1 x x Pmax) The probabilities control the search Regarding accuracy: avoid small values close to root Entire search branches disregarded more variance Regarding speed: avoid high values close to root More search branches higher execution time Some results We can obtain ~90% of accuracy using ~20% of time 44
45 Parallelism Subgraph Discovery is a tree shaped computation Search nodes are independent from each other {0,1,3} If we know where we are, we can continue from there Tree Nodes > Work Units 45
46 Parallel Problem Input: set of work units GTrie: (network, gtrie node, partial match) Goal: efficiently distribute work units among processors We need dynamic load balancing 46
47 Parallel approaches Environments we explored Clusters (distributed memory using MPI) [almost linear speedup up to 128 cores] Multicores (shared memory using threads) [almost linear speedup up to 64 cores] Some of the techniques Receiver initiated work stealing Diagonal splitting and efficient snapshots Random polling 47
48 Example Computation GTrie Node Graph Vertex 48
49 Example Computation GTrie Node Graph Vertex 49
50 Example Computation GTrie Node Graph Vertex 50
51 Example Computation GTrie Node Graph Vertex 51
52 Example Computation GTrie Node Graph Vertex Current Work Unit Explored Work Units STOP 52
53 Example Computation Keep Give to requester Both Diagonal Splitting 53
54 Some publications Core sequential algorithms  GTries: a data structure for storing and finding subgraphs. Data Mining and Know. Discovery, Towards a faster networkcentric subgraph census. ASONAM' Querying Subgraph Sets with GTries. DBSocial'2012 (best paper award)  Strategies for Network Motifs Discovery. EScience Sampling approach  Efficient Subgraph Frequency Estimation with GTries. WABI' RandFaSE: fast approximate subgraph census. SNAM, 2015 Parallel approach  Parallel subgraph counting for multicore architectures. ISPA' A Scalable Parallel Approach for Subgraph Census Computation. MuCoCos' Parallel Discovery of Network Motifs. Journal of Parallel and Distributed Computing Efficient Parallel Subgraph Counting using GTries. IEEE Cluster'2010. Concept variation and applications  Discovering Colored Network Motifs. Complenet' Motif Mining in Weighted Networks. Damnet'2012 (and new related in paper ACMSAC'2015)  Coauthorship network comparison across research fields using motifs. ASONAM'
55 Software Reference sequential implementation is available online and uses C++: More versions available by request and we are working on giving more options to practitioners (ex: GUI, plugin for Gephi/Cytoscape) Graphlets, large scale and adaptive sampling almost public 55
56 Complex Networks "Group Pedro Ribeiro Fernando Silva (leader) David Aparício (PhD Student) Multicores, Graphlets Pedro Paredes (BSc Student) NetworkCentric Graph Theory (leader) Miguel Araújo S. Choobdar (PhD Student) (former PhD Stud) Community Detection Matrix Factorization Structural Roles Temporal, Weigths Miguel Silva (MSc Student) Random Networks Ahmad Naser (MSc Student) GUI and Visualization Some collaborations (ongoing and/or starting):  M. Kaiser (U Newcastle) [Brain Networks]  L. F. Costa (U São Paulo) [Null Models]  K. Pingali (UT Austin) [Temporal Patterns]  C. Faloutsos (CMU) [Large Scale Networks]  S. Parthasarathy (Ohio State) [Roles] 56
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